On extension of polynomial functions
type: Item
instance of: scholarly article
title: On extension of polynomial functions
zbMATH Open document ID: 0829.39006
DOI: 10.1007/BF03323050
author: Roman Ger
published in: Results in Mathematics
publication date: 1996-01-25T00:00:00Z
review text: The author studies the solutions of the functional equation \[ \sum^{n+1}_{j=0} (-1)^{n+1 -j} {n+1\choose j} f\Bigl(\bigl( 1- {j\over n+1}\bigr) x+{j\over n+1} y\Bigr)=0, \] where the unknown function \(f\) is a mapping from a \(Q\)-convex subset \(D\) of a linear space \(X\) over the field of rationals into another linear space \(Y\) over \(Q\). By means of some elegant new tools it is proved that any such \(f\) admits an extension \(F:X\to Y\) of the form \(F(x)=A^0+A^1(x)+\cdots+A^n(x)\), where \(A^0\in Y\) and the maps \(A_k:X^k\to Y\) such that \(A_k(x,\ldots, x)=A^k(x)\) are \(k\)-additive and symmetric. The uniqueness problem of the extension \(F\) is discussed.
reviewed by: Claudi Alsina
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 46S10
zbMATH DE Number: 785172
zbMATH Keywords: polynomial functions
zbMATH Keywords: \(Q\)-linear spaces
zbMATH Keywords: \(Q\)-convexity
zbMATH Keywords: functional equation
zbMATH Keywords: uniqueness
zbMATH Keywords: extension
MaRDI profile type: Publication
cites work: Boundedness and continuity of additive and convex functionals
cites work: On vanishing n-th ordered differences and Hamel bases
cites work: Q3327052
cites work: Q4002300
description: scientific article; zbMATH DE number 785172